Express 0.78 as a fraction a/b where a,b ∈ Z Expand and Simplify: (7y2 – 6y – 8)(−2y + 2) Factorise the…

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Express 0.78 as a fraction a/b where a,b ∈ Z Expand and Simplify: (7y2 – 6y – 8)(−2y + 2) Factorise the following expressions fully. 3.1 8x2 - 6x + 1 3.2 125x3 + 1

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1.1 Express $0.7\overline{8}$ as a fraction $\frac{a}{b}$ where $a, b \in \mathbb{Z}$

Let $x$ be the repeating decimal.
$x = 0.7\overline{8}$
Multiply by powers of 10 to isolate the repeating part.
$10x = 7.\overline{8} \quad (1)$ $100x = 78.\overline{8} \quad (2)$
Subtract equation (1) from equation (2).
$100x - 10x = 78.\overline{8} - 7.\overline{8}$ $90x = 71$
Solve for $x$ and simplify the fraction.
$x = \frac{71}{90}$
The fraction is $\boxed{\frac{71}{90}}$ .

1.2 Expand and Simplify: $(7y^2 - 6y - 8)(-2y + 2)$
Distribute each term from the first polynomial to each term in the second polynomial.
$= 7y^2(-2y) + 7y^2(2) - 6y(-2y) - 6y(2) - 8(-2y) - 8(2)$
Perform the multiplications.
$= -14y^3 + 14y^2 + 12y^2 - 12y + 16y - 16$
Combine like terms.
$= -14y^3 + (14y^2 + 12y^2) + (-12y + 16y) - 16$ $= -14y^3 + 26y^2 + 4y - 16$
The expanded and simplified expression is $\boxed{-14y^3 + 26y^2 + 4y - 16}$ .

1.3 Factorise the following expressions fully.

1.3.1 $8x^2 - 6x + 1$
Find two numbers that multiply to $8 \times 1 = 8$ and add up to $-6$ .
These numbers are $-2$ and $-4$ .
Rewrite the middle term using these two numbers.
$8x^2 - 2x - 4x + 1$
Factor by grouping.
$2x(4x - 1) - 1(4x - 1)$
Factor out the common binomial factor $(4x - 1)$ .
$(2x - 1)(4x - 1)$
The factorised expression is $\boxed{(2x - 1)(4x - 1)}$ .

1.3.2 $125x^3 + 1$
Recognize this as a sum of cubes, which follows the formula $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$ .
Identify $a$ and $b$ .
$a^3 = 125x^3 \implies a = 5x$ $b^3 = 1 \implies b = 1$
Substitute $a$ and $b$ into the sum of cubes formula.
$(5x + 1)((5x)^2 - (5x)(1) + (1)^2)$
Simplify the terms.
$(5x + 1)(25x^2 - 5x + 1)$
The factorised expression is $\boxed{(5x + 1)(25x^2 - 5x + 1)}$ .

Express 0.78 as a fraction a/b where a,b ∈ Z Expand and Simplify: (7y2 – 6y – 8)(-2y + 2) Factorise the following expressions fully. 3.1 8x2 - 6x + 1 3.2 125x3 + 1

Step-by-step solution

Let $x$ be the repeating decimal.

Multiply by powers of 10 to isolate the repeating part.

Subtract equation (1) from equation (2).

Solve for $x$ and simplify the fraction.

Distribute each term from the first polynomial to each term in the second polynomial.

Perform the multiplications.

Combine like terms.

Find two numbers that multiply to $8 \times 1 = 8$ and add up to $-6$ .

Rewrite the middle term using these two numbers.

Factor by grouping.

Factor out the common binomial factor $(4x - 1)$ .

Recognize this as a sum of cubes, which follows the formula $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$ .

Identify $a$ and $b$ .

Substitute $a$ and $b$ into the sum of cubes formula.

Simplify the terms.

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